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SSC CGL 2024 PYQ — Question 54

SSC CGL - 19 Sep 2024 - Shift 3 • Quantitative Aptitude • Q54

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If A is an acute angle, then 1−cosA1+cosA+1+cosA1−cosA\sqrt{1−cosA\over1+cosA}+\sqrt{1+cosA\over 1−cosA} is equal to :

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Correct answer: A

Explanation

The correct answer is option 1 i.e. 2 cosecA.

1−cosA1+cosA+1+cosA1−cosA\sqrt{\frac{1−cosA}{1+cosA}}+\sqrt{\frac{1+cosA}{1-cosA}}

=(1−cosA)(1−cosA)(1+cosA)(1−cosA)+(1+cosA)(1+cosA)(1−cosA)(1+cosA)=\sqrt{\frac{(1−cosA)(1−cosA)}{(1+cosA)(1−cosA)}}+\sqrt{\frac{(1+cosA)(1+cosA)}{(1-cosA)(1+cosA)}}

=(1−cosA)21−cos2A+(1+cosA)21−cos2A=\sqrt{\frac{(1−cosA)^2}{1-cos^2A}}+\sqrt{\frac{(1+cosA)^2}{1-cos^2A}}

=(1−cosA)2sin2A+(1+cosA)2sin2A=\sqrt{\frac{(1−cosA)^2}{sin^2A}}+\sqrt{\frac{(1+cosA)^2}{sin^2A}}   [∵ sin2A = 1 - cos2A]

=1−cosAsinA+1+cosAsinA=\frac{1-cosA}{sinA}+\frac{1+cosA}{sinA}

= 2/sinA = 2cosecA